Problem: Evaluate the iterated integral. $ \int_0^\pi \left( \int_0^{\frac{\pi}{2}} \cos(x)\sin(y) \, dx \right) dy =$ Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{3}{2}$ (Choice B) B $2$ (Choice C) C $-1$ (Choice D) D $0$
Solution: Evaluate the inner integral: $\begin{aligned} \int_0^\pi \left( \int_0^{\frac{\pi}{2}} \cos(x)\sin(y) \, dx \right) dy &= \int_0^\pi \left[ \sin(x)\sin(y) \right]_0^{\frac{\pi}{2}} \, dy \\ \\ &= \int_0^\pi \sin(y) \, dy \end{aligned}$ Evaluate the outer integral: $\begin{aligned} \int_0^\pi \sin(y) \, dy &= -\cos(y) \bigg|_0^\pi \\ \\ &= 2 \end{aligned}$ The answer: $ \int_0^\pi \left( \int_0^{\frac{\pi}{2}} \cos(x)\sin(y) \, dx \right) dy = 2$